Unique existence results of solutions are known according to
several formulations of initial boundary value problems.
In Sakamoto and Yamamoto [1]}, in the case of
symmetric \(A\)  where \(b_j=0\)  for \(1\le j \le d\)  in (1.1),
the unique existence is proved by means of the Fourier method, but the class of
solutions is not the same as here.
In the case where \(b_j\)  for \(j=1,..., d\)  are not necessarily zero,
assuming that the initial value \(a\)  is zero, Theorem 6.1 is proved in
Gorenflo, Luchko and Yamamoto [2]}. In both [2]} and
[1]}, it is assumed that all the coefficients are independent of \(t\)  and
and \(c=c(x) \le 0\)  for \(x\in \Omega \) .
The work Kubica, Ryszewska and Yamamoto [5]} proved Theorem 6.1 in a
general case where \(a_{ij}, b_j, c\)  depends both on \(x\)  and \(t\)  without
extra assumpion \(c\le 0\) . In other words, Theorem 6.1 is a special case of
Theorem 4.2 in [5]}.
Furthermore, there have been other works on the well-posedness for
initial boundary value problems and we are
restricted to some of them: Bajlekova [7]}, Kubica and Yamamoto
[8]}, Luchko [9]}, [10]}, Luchko and Yamamoto [11]},
Zacher [12]}. See also Prüss [13]} for a monograph on related
integral
equations, and a recent book Jin [14]} which mainly studies the symmetric
\(A\) . As for further references up to 2019, the
handbooks [15]} edited by A.Kochubei and Y. Luchko
are helpful. Most of the above works discuss the case of
\(F \in L^2(0,T;L^2(\Omega ))\) .
